Equivalence of Three Different Kinds of Optimal Control Problems for Heat Equations and Its Applications

Wang, Gengsheng; Xu, Yashan

doi:10.1137/110852449

Cited by 33 publications

(31 citation statements)

References 15 publications

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“…Hence, instead of solving the time-optimal control problem directly, we can search for a root of the δ(·)-value function. A similar equivalence first appeared in [35] (see also [34,Section 5.4]) for the situation where one aims at delaying the activation of the control as long as possible. However, to the best of our knowledge it has never been considered for an algorithmic approach.…”

Section: Introductionmentioning

confidence: 56%

Time-optimality by distance-optimality for parabolic control systems

Bonifacius

Kunisch

2020

ESAIM: M2AN

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Section: Introductionmentioning

confidence: 56%

Time-optimality by distance-optimality for parabolic control systems

Bonifacius

Kunisch

2020

ESAIM: M2AN

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“…More importantly, with the aid of Theorem 1.1, we can build up the bang-bang property of time optimal control problems for the above-mentioned controlled equations. This property is extremely important in the studies of time optimal control problems (cf., e.g., [20], [21], [27], [30], [39], [40], [41]). These applications will be presented in Section 3 of this paper.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Observability Inequalities from Measurable Sets for Some Abstract Evolution Equations

Wang¹,

Zhang²

2017

SIAM J. Control Optim.

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In this paper, we build up two observability inequalities from measurable sets in time for some evolution equations in Hilbert spaces from two different settings. The equation reads: u ′ = Au, t > 0, and the observation operator is denoted by B. In the first setting, we assume that A generates an analytic semigroup, B is an admissible observation operator for this semigroup (cf. [36]), and the pair (A, B) verifies some observability inequality from time intervals. With the help of the propagation estimate of analytic functions (cf. [35]) and a telescoping series method provided in the current paper, we establish an observability inequality from measurable sets in time. In the second setting, we suppose that A generates a C 0 semigroup, B is a linear and bounded operator, and the pair (A, B) verifies some spectral-like condition. With the aid of methods developed in [2] and [29] respectively, we first obtain an interpolation inequality at one time, and then derive an observability inequality from measurable sets in time. These two observability inequalities are applied to get the bang-bang property for some time optimal control problems.Key words. Evolution equations in Hilbert spaces, observability inequality in measurable sets, telescoping series method, propagation estimate of analytic functions, bang-bang property of time optimal controls AMS Subject Classifications. 93B07, 93C25

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“…We refer to [15,41] for semi-discrete finite element approximations, and [34,43] for perturbations of equations. About more works on time optimal control problems, we would like to mention [2,10,11,16,17,18,21,22,25,27,30,31,35,37,38,39,40,42,44] and the references therein.…”

Section: Resultsmentioning

confidence: 99%

Time optimal sampled-data controls for the heat equation

Wang

Yang

Zhang

2017

Comptes Rendus. Mathématique

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In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls. Our design is reasonable from perspective of sampled-data controls. And it might provide a right way for the numerical approach of a time optimal distributed control problem, via the corresponding semi-discretized (in time variable) time optimal control problem.The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem. And obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but also prove that such estimates are optimal in some sense.

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Equivalence of Three Different Kinds of Optimal Control Problems for Heat Equations and Its Applications

Cited by 33 publications

References 15 publications

Time-optimality by distance-optimality for parabolic control systems

Time-optimality by distance-optimality for parabolic control systems

Observability Inequalities from Measurable Sets for Some Abstract Evolution Equations

Time optimal sampled-data controls for the heat equation

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